Chapter PII, Problem 10RE

(a)

To determine

To explain what did the instructor predict for her score on the final if Susan scored a $70$ on the midterm.

Answer to Problem 10RE

The final score is $73$ .

Explanation of Solution

It is given in the question that a statistics instructor created a linear regression equation to predict students’ final exam score from their midterm exam scores. The regression equation is:

  

Now, if Susan scored a $70$ on the midterm then the final score of her exam will be as:

  

(b)

To determine

To explain how big is her residual.

Answer to Problem 10RE

The residual is $7$ .

Explanation of Solution

It is given in the question that a statistics instructor created a linear regression equation to predict students’ final exam score from their midterm exam scores. The regression equation is:

  

Now, if Susan got an $80$ on the final, then the difference will be:

  $\begin{array}{l}=80-73\\ =7\end{array}$

(c)

To determine

To find out what is the correlation between the two tests.

Answer to Problem 10RE

  $r=0.75$ .

Explanation of Solution

It is given in the question that a statistics instructor created a linear regression equation to predict students’ final exam score from their midterm exam scores. The regression equation is:

  

Now, we know the formula for to calculate the slope and we know the standard deviation of the final was $12$ points and the standard deviation of the midterm was $10$ points. Thus, we have,

  $\begin{array}{l}b=r\frac{s_x}{s_y}\\ \Rightarrow 0.9=r\frac{12}{10}\\ \Rightarrow 9=r12\\ \Rightarrow r=0.75\end{array}$

(d)

To determine

To explain how many points would someone need to score on the midterm to have a predicted final score of $100$ .

Answer to Problem 10RE

  $100$ points someone would need to score on the midterm to have a predicted final score of $100$ .

Explanation of Solution

It is given in the question that a statistics instructor created a linear regression equation to predict students’ final exam score from their midterm exam scores. The regression equation is:

  

Thus, the points that someone would need to score on the midterm to have a predicted final score of $100$ is calculated as:

  

So, $100$ points someone would need to score on the midterm to have a predicted final score of $100$ .

(e)

To determine

To explain why you cannot estimate this student’s midterm score from the information given.

Explanation of Solution

It is given in the question that a statistics instructor created a linear regression equation to predict students’ final exam score from their midterm exam scores. The regression equation is:

  

Thus, we cannot estimate this student’s midterm score from the information given because the model is made for finding the final score not the inverse.

(f)

To determine

To find out what is the residual for this student.

Answer to Problem 10RE

The residual is $85$ .

Explanation of Solution

It is given in the question that a statistics instructor created a linear regression equation to predict students’ final exam score from their midterm exam scores. The regression equation is:

  

Thus, one of the student scored $100$ on the midterm but scored only $15$ in the final exam. Thus, predicted final score for the student is:

  

So, the residual is:

  $\begin{array}{l}=100-15\\ =85\end{array}$

(g)

To determine

To explain will the $R^2$ of the regression increase, decrease or remain the same.

Answer to Problem 10RE

It will increase.

Explanation of Solution

It is given in the question that a statistics instructor created a linear regression equation to predict students’ final exam score from their midterm exam scores. The regression equation is:

  

Thus, $R^2$ of the regression will increase because the model will be more closely related to the rest of the values.

(h)

To determine

To explain will the slope of the new line increase or decrease.

Answer to Problem 10RE

The slope will increase.

Explanation of Solution

It is given in the question that a statistics instructor created a linear regression equation to predict students’ final exam score from their midterm exam scores. The regression equation is:

  

Thus, the slope of the new line increases because the rest of the final test scores are higher compared to the midterm scores.